3) A more general sequence, as in 1), but of different objects and not necessarily numbers, frequently displayed as

or likewise a sequence of functions

In all these cases, of course, none of n, m, or p are known: they are just symbols, or algebraic expressions, representing integer values.

These most typical cases of symbolic sequences have been implemented in Maple since day 1 using the `$` operator. Cases 1), 2) and 3) above are respectively entered as , , and or To have computer algebra representations for all these symbolic sequences is something wonderful, I would say unique in Maple.

Until recently, however, the typesetting of these symbolic sequences was frankly poor, input like or ` just being echoed in the display. More relevant: too little could be done with these objects; the rest of Maple didn't know how to add, multiply, differentiate or map an operation over the elements of the sequence, nor for instance count the sequence's number of elements.

All this has now been implemented. What follows is a brief illustration.

>

First of all, now these three types of sequences have textbook-like typesetting:

>

(1)

>

(2)

For the above, a$p works the same way

>

(3)

Moreover, this now permits textbook display of mathematical functions that depend on sequences of paramateters, for example:

>

(4)

>

(5)

More interestingly, these new developments now permit differentiating these functions even when their arguments are symbolic sequences, and displaying the result as in textbooks, with copy and paste working properly, for instance

>

(6)

It is very interesting how much this enhances the representation capabilities; to mention but one, this makes 100% possible the implementation of the Faa-di-Bruno formula for the nth symbolic derivative of composite functions (more on this in a post to follow this one).

But the bread-and-butter first: the new package for handling sequences is

>

(7)

The five commands that got loaded do what their name tells. Consider for instance the first kind of sequences mentione above, i.e

>

(8)

Check what is behind this nice typesetting

>

`$`(n .. m)

All OK. How many operands (an abstract version of Maple's nops command):

>

(9)

That was easy, ok. Add the sequence

>

(10)

Multiply the sequence

>

(11)

Map an operation over the elements of the sequence

>

(12)

>

`$`(f(j), j = n .. m)

Map works as map, i.e. you can map extra arguments as well

>

(13)

All this works the same way with symbolic sequences of forms , and . For example:

>

(14)

>

`$`(a, p)

>

(15)

>

(16)

>

(17)

Differentation also works

>

(18)

>

(19)

>

(20)

For a symbolic sequence of type 3)

>

(21)

>

(22)

>

(23)

>

(24)

The following is nontrivial: differentiating the sequence , with respect to should return 1 when (i.e the running index has the value k), and 0 otherwise, and the same regarding m and k. That is how it works now:

>

(25)

>

`$`(piecewise(k = i, 1, 0), i = n .. m)

>

(26)

>

(27)

>

`$`((diff(f(a[i]), a[i]))*piecewise(k = i, 1, 0), i = n .. m)

And that is it. Summarizing: in addition to the former implementation of symbolic sequences, we now have textbook-like typesetting for them, and more important: Add, Multiply, Differentiate, Map and Nops. :)

The first large application we have been working on taking advantage of this is symbolic differentiation, with very nice results; I will see to summarize them in a post to follow in a couple of days.

I am writing here because of a problem with writing mathematical expressions in Maple T.A.. I have been using it since two years, and I have a lot if questions created in version 9.0 and 9.5. A few months ago my university installed T.A. 10. At the beginning there were problems with the connections between T.A. and the Maple server. After the administrators got over these, there were another problems.

There are a lot of questions where I use greek letters, for example Sigma. Earlier it was easy, I wrote

in the 'Algorithm' section, and I could write

in the Text of the question. The letter had a perfect italic and bold style, like as it would be created with Equation Editor.

I am attempting to maximize Planck's intesnity function for any given temperature T with respect to lambda. The function is as follows:

My initial attempt was to simply take the derivative of this function with respect to labda, set it equal to zero, and then solve for lamda. Theoretically this should give me the value of lambda that maximizes the intesity with respect to the constants / values h, c, k, and T (where T is temperature).

However doing this in maple just leads to the result "warning, solutions may have been lost".

HiTwo new things recently added to the latest version of Physics available on Maplesoft's R&D Physics webpage are worth mentioning outside the framework of Physics.

automaticsimplification. This means that after "Physics:-Setup(automaticsimplification=true)", the output corresponding to every single input (literally) gets automatically simplified in size before being returned to the screen. This is fantastically convenient for interactive work in most situations.

Add Physics:-Library:-Assume, to perform the same operations one typically performs with the assume command, but without the side effect that the variables get redefined. So the variables do not get redefined, they only receive assumptions.

This new Assume implements the concept of an "extended assuming". It permits re-using expressions involving the variables being assumed, expressions that were entered before the assumptions were placed, as well as reusing all the expressions computed while the variables had assumptions, even after removing the variable's assumptions. None of this is possible when placing assumptions using the standard assume. The new routine also permits placing assumptions on global variables that have special meaning, that cannot be redefined, e.g. the cartesian, cylindrical or spherical coordinates sets, or the coordinates of a coordinate spacetime system within the Physics package, etc.

Examples:

>

This is Physics from today:

>

(1.1)

•

Automatic simplification is here. At this point automaticsimplification is OFF by default.

>

(1.2)

Hence, for instance, if you input the following expression, the computer just echoes your input:

>

(1.3)

There is however some structure behind (1.3) and, in most situations, it is convenient to have these structures apparent, in part because they frequently provide hints on how to proceed ahead, but also because a more compact expression is, roughly speaking, simpler to understand. To see this automaticsimplification in action, turn it ON:

>

(1.4)

Recall this same expression (you could input it with the equation label (1.3) as well)

>

(1.5)

What happened: this output, as everything else after you set and with no exceptions, is now further processed with simplify/size before being returned. And enjoy computing with frankly shorter expressions all around! And no need anymore for "simplify(%, size)" every three or four input lines.

Another example, typical in computer algebra where expressions become uncomfortably large and difficult to read: convert the following input to 2D math input mode first, in order to compare what is being entered with the automatically simplified output on the screen

>

(1.6)

You can turn automaticsimplification OFF the same way

>

(1.7)

•

New facility; welcome to the world of "extended assuming" :)

Consider a generic variable, x. Nothing is known about it

>

x:

nothing known about this object

Each variable has associated a number that depends on the session, and the computer (internally) uses this number to refer to the variable.

>

(1.8)

When using the assume command to place assumptions on a variable, this number, associated to it, changes, for example:

>

>

(1.9)

Indeed, the variable x got redefined and renamed, it is not anymore the variable x referenced in (1.8).

>

Originally x, renamed x~:

is assumed to be: RealRange(Open(0),Open(1/2*Pi))

The semantics may seem confusing but that is what happened, you enter x and the computer thinks x~, not xanymore.This means two things:

1) all the equations/expressions, entered before placing the assumptions on x using assume, involve a variable xthat is different than the one that exists after placing the assumptions, and so these previous expressions cannot be reused. They involve a different variable.

2) Also, because, after placing the assumptions using assume, x refers to a different object, programs that depend on the x that existed before placing the assumptions will not recognize the new x redefined by assume.

For example, if x was part of a coordinate system and the spacetime metric depends on it, the new variable xredefined within assume, being a different symbol, will not be recognized as part of the dependency of This posed constant obstacles to working with curved spacetimes that depend on parameters or on coordinates that have a restricted range. These problems are resolved entirely with this new Library:-Assume, because it does not redefine the variables. It only places assumptions on them, and in this sense it works like assuming, not assume. As another example, all the Physics:-Vectors commands look for the cartesian, cylindrical or spherical coordinates sets in order to determine how to proceed, but these variables disappear if you use assume to place assumptions on them. For that reason, only assuming was fully compatible with Physics, not assume.

To undo assumptions placed using the assume command one reassigns the variable x to itself:

>

(1.10)

Check the numerical address: it is again equal to (1.8)

>

(1.11)

·All these issues get resolved with the new Library:-Assume, that uses all the implementation of the existingassume command but with a different approach: the variables being assumed do not get redefined, and hence: a) you can reuse expressions/equations entered before placing the assumptions, you can also undo the assumptions and reuse results obtained with assumptions. This is the concept of an extended assuming. Also, commands that depend on these assumed variables will all continue to work normally, before, during or after placing the assumption, because the variables do not get redefined.

Example:

>

x:

nothing known about this object

So this simplification attempt accomplishes nothing

>

(1.12)

Let's assume now that

>

(1.13)

The new command echoes the internal format representing the assumption placed.

a) The address is still the same as (1.8)

>

(1.14)

So the variable did not get redefined. The system however knows about the assumption - all the machinery of the assume command is being used

>

Originally x, renamed x:

is assumed to be: RealRange(Open(0),Open(1/2*Pi))

Note that the renaming is to the variable itself - i.e. no renaming.

Hence, expressions entered before placing assumptions can be reused. For example, for (1.12), we now have

>

(1.15)

To clear the assumptions on x, you can use either of Library:-Assume(x=x) or Library:-Assume(clear = {x, ...}) in the case of many variables being cleared in one go, or in the case of a single variable being cleared:

>

>

x:

nothing known about this object

The implementation includes the additionally functionality, for that purpose add the keyword additionallyanywhere in the calling sequence. For example:

>

(1.16)

>

Originally x, renamed x:

is assumed to be: RealRange(Open(0),infinity)

>

(1.17)

>

In summary, the new Library:-Assume command implements the concept of an extended assuming, that can be turned ON and OFF at will at any moment without changing the variables involved.

I am using Maple 12 in Win 64. I have some difficulties in evaluating expressions in Maple. For example simply writing cos(x) then differentiating it w.r.t x. After writing cos(x) in 2D Math mode, I do right click for selecting diff but it does not work. The pointer turns in to busy mode and nothing comes to select. Also, I tried this operation in the tutorial file teaching differentiation, however problem persists.

I have some lengthy formulas in the maple. I don't want to waste time on rewritting them in a word document.Is there a way to import those equations in a clean and tidy form to a word document or the mathtype program or something else! :)

Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on an upcoming webinar we think may be of interest to the MaplePrimes community. For the complete list of upcoming webinars, visit our website.

Hollywood Math 2

In this second installment of the Hollywood Math webinar series, we will present some more examples of mathematics being used in Hollywood films and popular hit TV series. For instance, have you wondered how Ben Campbell solved his professor’s challenge so easily in the movie “21”? Or about the details of the Nash equilibrium that John Nash first developed in a “A Beautiful Mind”? We’ve got the answers! These relevant, and exciting examples can be used as material to engage your students with examples familiar to them, or you can just attend the webinar for its entertainment value.

Anyone with an interest in mathematics, especially high school and early college math educators, will be both entertained and informed by attending this webinar. At the end of the webinar you’ll be given an opportunity to download an application containing all of the examples that we demonstrate.